arXiv subjects

Nuh Aydin

Publications and source records attributed to Nuh Aydin.

At least 19 recordsLinked to original sources

arXiv2026

Resolution of an Open Problem on Quasi-Cyclic Codes over $\mathbb{Z}_4$ and New Quaternary Linear Codes

Given a cyclic code $C_g = \langle g(x) \rangle$ of odd length $m$ over $\mathbb{Z}_4$, one common way to build a quasi-cyclic (QC) code is to pick $f_1, \dots, f_\ell \in \mathbb{Z}_4[x]$ and let $C = \langle (f_1 g, \dots, f_\ell g) \rangle$. Because $\mathbb{Z}_4$ is not a field, the type of $C_g$ ($4^{k_1}2^{k_2}$) is not necessarily inherited by $C$. Determining conditions under which the type of $C_g$ is inherited by $C$ was posed as an open problem recently in \cite{AydinLuOnta2023}. In this paper, we settle this problem. We first give two sufficient conditions for the type to be preserved: one requires a single $f_i$ to be coprime to $x^m-1$ over $\mathbb{F}_2$, the other only requires the $f_i$ to be jointly coprime to it. Neither condition is necessary in general. Using the fact that $x^m-1$ is squarefree for odd $m$, we decompose $\mathbb{Z}_4[x]/\langle x^m-1\rangle$ into a product of finite chain rings using Chinese remainder theorem (CRT) and derive a condition on $f_1, \dots, f_\ell$ that is both necessary and sufficient for $C$ to match the type of $C_g$. This condition depends only on the irreducible factors of $x^m-1$ where $g$ does not already vanish. This also yields a simple test for when $C$ is a free $\mathbb{Z}_4$-module. Finally, we report many new QC codes over $\mathbb{Z}_4$, found by computer searches using Magma software~\cite{Magma1997} guided by this criterion, with Lee distances greater than previously known codes of the same type.

cs.IT↗

arXiv2026

Bipartite Extremal Numbers of Trees

We study a restriction of the classical Erdős--Sós problem, the extremal number of trees, to the class of bipartite host graphs, both when only the order of the host is prescribed and when its two part-sizes are fixed. We give natural lower-bound constructions and formulate corresponding linear upper-bound conjectures. We apply a weighted variant of $k$-minimality to prove upper bounds for a broad family of trees including brooms, trees with part-sizes obeying certain inequalities, and all trees on at most seven vertices, resolving part of a problem of Caro, Patkós and Tuza up to additive constants. We also relate the fixed-part extremal number of a tree to the ordinary extremal number, and consider an oriented bipartite extremal function analogous to the Zarankiewicz function.

math.CO↗

arXiv2026

On the History of the Square and Multiply Algorithm

The square-and-multiply algorithm, also known as binary exponentiation or repeated squaring, is a standard method for fast exponentiation in modern computation. Its historical origins, however, remain uncertain. This paper examines the emergence and progressive formalization of the method through selected primary sources. Particular attention is given to Jamshid al-Kashi's fifteenth-century Miftah al-Hisab, where the procedure is presented explicitly as a general computational method and claimed by al-Kashi as his own innovation. Earlier instances of successive squaring are identified in the works of al-Uqlidisi and al-Biruni, although in these cases the technique appears in particular calculations rather than as a fully articulated general rule. The earliest known antecedent is found in Pingala's prosodic studies in ancient India (c. 200 BCE), which seem to presuppose the conceptual basis of the method in their use of binary representation. As part of the historical development, Legendre's 1798 worked example is one of the earliest documented European use of the algorithm which appears as a subordinate step within a specific number-theoretic computation. The evidence suggests not a single continuous line of transmission, but the repeated independent reappearance of related procedures in distinct contexts. By the twentieth century, square-and-multiply became a special case within the broader theory of addition chains. By exploring this intellectual progression, this paper sheds some light on the historical background of an algorithm that is prominent in modern computation.

math.HO↗

arXiv2026

Combinatorial Bounds for Codes over Metric Spaces: Ramsey-Sidorenko Thresholds and Subgraph Counts

This paper investigates the relationship between coding theory and extremal combinatorics by representing codes in general metric spaces as independent sets in proximity graphs. We provide a generalized framework for the Gilbert-Varshamov (GV) bound applicable to codes over any finite metric space and explore the conditions under which global combinatorial parameters can force the existence of codes exceeding this bound. Central to our analysis is the introduction of Ramsey-Sidorenko and independence-forcing graphs. We establish density thresholds for various graph families and utilize the Karush--Kuhn--Tucker conditions to analyze entropy optimization in the Hamming case. Furthermore, we derive upper bounds on code sizes using fractional packings in vertex-transitive and nonedge-transitive graphs. Our findings demonstrate that local subgraph statistics alone are insufficient to surpass the GV bound in the Hamming case, suggesting that improvements must stem from large-scale structural properties of the space.

math.CO↗

arXiv2026

Gap-Sums via Quasi-Arithmetic Means with Applications to Fibonacci and Lucas Sequences

We develop a unified framework for studying the integers missing between consecutive terms of an increasing integer sequence, extending Barry's arithmetic gap-sum to geometric and harmonic analogues via the theory of quasi-arithmetic means. All three gap-sums admit a common interpretation: each equals the gap size multiplied by the appropriate mean of the missing integers. Building on this, we prove a general sparse summation theorem expressing the sum of a strictly monotonic function over a sparse integer sequence as the full range sum corrected by the gap-sums of the missing portions. Specializing on the three Pythagorean means recovers a classical formula of al-Kāsh\=ı from the fifteenth century in the arithmetic case, and yields explicit formulas in the geometric and harmonic cases. As a concrete application of the geometric case, we derive a product identity involving the Fuss--Catalan numbers. Applying the harmonic case to the Fibonacci and Lucas sequences, we establish that the harmonic gap-sum converges to $\ln(α)$ exponentially, where $α$ is the golden ratio, and derive explicit two-term asymptotic expansions for the tails of the reciprocal Fibonacci and Lucas series with closed-form coefficients, and establish the asymptotic formula $H_{u_n} \sim n\ln(α)$ for both $u_n = F_n$ and $u_n = L_n$, with explicit $O(1)$ error terms that differ due to their distinct initial conditions. As a further consequence, by comparing the gap-sum expansions with the classical Hardy--Wright expansion of harmonic numbers, we derive exact series identities expressing Euler's constant $γ$ in terms of harmonic numbers at Fibonacci and Lucas indices, and obtain a new identity relating the reciprocal Fibonacci constant $ψ$ and the reciprocal Lucas constant $ψ_L$.

math.NT↗

arXiv2026

New Codes from Cyclic and Negacyclic Codes of Even Length over $\mathbb{Z}_4$

This paper uses theoretical results previously established in the literature to design search algorithms to find new linear codes over $\mathbb{Z}_4$ from cyclic and negacyclic codes of even length. As a result of these searches, we have found 2500 new cyclic codes and 730 negacyclic codes. These new codes exhibit improved parameters compared to previously known codes. Additionally, we have obtained binary quantum codes with good parameters from such $\mathbb{Z}_4$ codes.

cs.IT↗

arXiv2026

From Historical Puzzles to Grammatical Constraints: Circular Partitions, Generalized Run-Length Encodings, and Polynomial-Time Decidability

Motivated by a historical combinatorial problem that resembles the well-known Josephus problem, we investigate circular partition algorithms and formulate problems in deterministic finite automata with practical algorithms. The historical problem involves arranging individuals on a circle and eliminating every k-th person until a desired group remains. We analyze both removal and non-removal approaches to circular partitioning, establishing conditions for balanced partitions and providing explicit algorithms. We introduce generalized run-length encodings over partitioned alphabets to capture alternating letter patterns, computing their cardinalities using Stirling numbers of the second kind. Connecting these combinatorial structures to formal language theory, we formulate an existence problem: given a context-free grammar over a dictionary and block-pattern constraints on letters, does a valid sentence exist? We prove decidability in polynomial time by showing block languages are regular and applying standard parsing techniques. Complete algorithms with complexity analysis are provided and validated through implementation on both historical and synthetic instances.

cs.FL↗

arXiv2025

Bounds and Equivalence of Skew Polycyclic Codes over Finite Fields

We study skew polycyclic codes over a finite field $\mathbb{F}_q$, associated with a skew polynomial $f(x) \in \mathbb{F}_q[x;σ]$, where $σ$ is an automorphism of $\mathbb{F}_q$. We start by proving the Roos-like bound for both the Hamming and the rank metric for this class of codes. Next, we focus on the Hamming and rank equivalence between two classes of polycyclic codes by introducing an equivalence relation and describing its equivalence classes. Finally, we present examples that illustrate applications of the theory developed in this paper.

cs.IT↗

arXiv2024

Elementary Constructions of Best Known Quantum Codes

Recently, many good quantum codes over various finite fields $F_q$ have been constructed from codes over extension rings or mixed alphabet rings via some version of a Gray map. We show that most of these codes can be obtained more directly from cyclic codes or their generalizations over $F_q$. Unless explicit benefits are demonstrated for the indirect approach, we believe that direct and more elementary methods should be preferred.

cs.IT↗

arXiv2023

On $(n,σ)-$equivalence relation between skew constacyclic codes

In this paper we generalize the notion of $n$-equivalence relation introduced by Chen et al. in \cite{Chen2014} to classify constacyclic codes of length $n$ over a finite field $\mathbb{F}_q$, where $q=p^r$ is a prime power, to the case of skew constacyclic codes without derivation. We call this relation $(n,σ)$-equivalence relation, where $n$ is the length of the code and $ σ$ is an automorphism of the finite field. We compute the number of $(n,σ)$-equivalence classes, and we give conditions on $ λ$ and $μ$ for which $(σ, λ)$-constacyclic codes and $(σ, λ)$-constacyclic codes are equivalent with respect to our $(n,σ)$-equivalence relation. Under some conditions on $n$ and $q$ we prove that skew constacyclic codes are equivalent to cyclic codes. We also prove that when $q$ is even and $σ$ is the Frobenius autmorphism, skew constacyclic codes of length $n$ are equivalent to cyclic codes when $\gcd(n,r)=1$. Finally we give some examples as applications of the theory developed here.

cs.IT↗

arXiv2023

A New Hybrid Cryptosystem Involving DNA,Rabin, One Time Pad and Fiestel

Information security is a crucial need in the modern world. Data security is a real concern, and many customers and organizations need to protect their sensitive information from unauthorized parties and attackers. In previous years, numerous cryptographic schemes have been proposed. DNA cryptography is a new and developing field that combines the computational and biological worlds. DNA cryptography is intriguing due to its high storage capacity, secure data transport, and massive parallel computing. In this paper, a new combination is proposed that offers good security by combining DNA, the Rabin algorithm, one time pad, and a structure inspired by Fiestel. This algorithm employs two keys. The first key is a DNA OTP key which is used for only one secure communication session. The second key, which combines the public and private keys, is a Rabin key. Additionally, by using a Feistel inspired scheme and randomness provided by DNA, the ciphertext is made harder to obtain without the private key.

cs.CR↗

arXiv2022

An Updated Database of $\mathbb{Z}_4$ Codes

Research on codes over finite rings has intensified since the discovery in 1994 of the fact that some best binary non-linear codes can be obtained as images of $\mathbb{Z}_4$-linear codes. Codes over many different finite rings has been a subject of much research in coding theory after this discovery. Many of these rings are extensions of $\mathbb{Z}_4$. As a result, an online database of $\mathbb{Z}_4$ was created in 2008. The URL of the original database on $\mathbb{Z}_4$ codes has recently changed. The purpose of this paper is to introduce the new, updated database of $\mathbb{Z}_4$ codes. We have made major updates to the database by adding 8701 new linear codes over $\mathbb{Z}_4$. These codes have been found through exhaustive computer searches on cyclic codes and by an implementation of the ASR search algorithm that has been remarkably fruitful to obtain new linear codes from the class of quasi-cyclic (QC) and quasi-twisted (QT) codes over finite fields. We made modifications to the ASR algorithm to make it work over $\mathbb{Z}_4$. The initial database contained few codes that were not free. We have added a large number of non-free codes. In fact, of the 8701 codes we have added, 7631 of them are non-free.

cs.IT↗

arXiv2021

Applications of Convolutional Codes to DNA Codes and Error-Correction

Convolutional codes are error-correcting linear codes that utilize shift registers to encode. These codes have an arbitrary block size and they can incorporate both past and current information bits. DNA codes represent DNA sequences and are defined as sets of words comprised of the alphabet A, C, T, G satisfying certain mathematical bounds and constraints. The application of convolutional code models to DNA codes is a growing field of biocomputation. As opposed to block codes, convolutional codes factor in nearby information bits, which makes them an optimal model for representing biological phenomena. This study explores the properties of both convolutional codes and DNA codes, as well as how convolutional codes are applied to DNA codes. It also proposes revisions to improve a current convolutional code model for DNA sequences.

cs.IT↗

arXiv2021

Additive Polycyclic Codes over $\mathbb{F}_{4}$ Induced by Binary Vectors and Some Optimal Codes

In this paper we study the structure and properties of additive right and left polycyclic codes induced by a binary vector $a$ in $\mathbb{F}_{2}^{n}.$ We find the generator polynomials and the cardinality of these codes. We also study different duals for these codes. In particular, we show that if $C$ is a right polycyclic code induced by a vector $a\in \mathbb{F}_{2}^{n}$, then the Hermitian dual of $C$ is a sequential code induced by $a.$ As an application of these codes, we present examples of additive right polycyclic codes over $\mathbb{F}_{4}$ with more codewords than comparable optimal linear codes as well as optimal binary linear codes and optimal quantum codes obtained from additive right polycyclic codes over $\mathbb{F}_{4}.$

cs.IT↗

arXiv2021

A Generalization of the ASR Search Algorithm to 2-Generator Quasi-Twisted Codes

One of the main goals of coding theory is to construct codes with best possible parameters and properties. A special class of codes called quasi-twisted (QT) codes is well-known to produce codes with good parameters. Most of the work on QT codes has been over the 1-generator case. In this work, we focus on 2-generator QT codes and generalize the ASR algorithm that has been very effective to produce new linear codes from 1-generator QT codes. Moreover, we also generalize a recent algorithm to test equivalence of cyclic codes to constacyclic codes. This algorithm makes the ASR search even more effective. As a result of implementing our algorithm, we have found 103 QT codes that are new among the class of QT codes. Additionally, most of these codes possess the following additional properties: a) they have the same parameters as best known linear codes, and b) many of the have additional desired properties such as being LCD and dual-containing. Further, we have also found a binary 2-generator QT code that is new (record breaking) among all binary linear codes and its extension yields another record breaking binary linear code.

cs.IT↗

arXiv2021

A Generalization of Cyclic Code Equivalence Algorithm to Constacyclic Codes

Recently, a new algorithm to test equivalence of two cyclic codes has been introduced which is efficient and produced useful results. In this work, we generalize this algorithm to constacyclic codes. As an application of the algorithm we found many constacyclic codes with good parameters and properties. In particular, we found 23 new codes that improve the minimum distances of BKLCs.

cs.IT↗

arXiv2021

New Binary and Ternary Quasi-Cyclic Codes with Good Properties

One of the most important and challenging problems in coding theory is to construct codes with best possible parameters and properties. The class of quasi-cyclic (QC) codes is known to be fertile to produce such codes. Focusing on QC codes over the binary field, we have found 113 binary QC codes that are new among the class of QC codes using an implementation of a fast cyclic partitioning algorithm and the highly effective ASR algorithm. Moreover, these codes have the following additional properties: a) they have the same parameters as best known linear codes, and b) many of the have additional desired properties such as being reversible, LCD, self-orthogonal or dual-containing. Additionally, we present an algorithm for the generation of new codes from QC codes using ConstructionX, and introduce 35 new record breaking linear codes produced from this method.

cs.IT↗

arXiv2021

A Database of Quantum Codes

Quantum error correcting codes (QECC) is becoming an increasingly important branch of coding theory. For classical block codes, a \href{codetables.de} {comprehensive database of best known codes} exists which is available online at \cite{codetables}. The same database contains data on best known quantum codes as well, but only for the binary field. There has been an increased interest in quantum codes over larger fields with many papers reporting such codes in the literature. However, to the best of our knowledge, there is no database of best known quantum codes for most fields. We established a new database of QECC that includes codes over $\mathbb{F}_{q^2}$ for $q\leq 29$. We also present several methods of constructing quantum codes from classical codes based on the CSS construction. We have found dozens of new quantum codes that improve the previously known parameters and also hundreds new quantum codes that did not exist in the literature.

cs.IT↗